TITLE: Axiom of Extensionality - Why not called equality?
QUESTION [5 upvotes]: Axiom of Extensionality, I understand that if two sets have exactly the same members they are equal. However why it is called Extensionality? Why not equality?

REPLY [6 votes]: "Extension" refers to a set being defined by its content, as opposed to "intension" which is a term to say it is defined by some form of specification.
Let A = {x in R such that -1 <= x <= +1}
Let B = {y in R such that y = sin(x) for some x in R}.
The sets have different intensions (specifications), but contain the same elements and therefore have the same extensions. So (according to the "Axiom of Extension") A = B.